-
NUMERICAL RANGE FOR RANDOM MATRICES
BENOÎT COLLINS1, PIOTR GAWRON2, ALEXANDER E. LITVAK3,KAROL
ŻYCZKOWSKI4,5
ABSTRACT. We analyze the numerical range of high-dimensional
ran-dom matrices, obtaining limit results and corresponding
quantitative es-timates in the non-limit case. For a large class of
random matrices theirnumerical range is shown to converge to a
disc. In particular, numericalrange of complex Ginibre matrix
almost surely converges to the disk ofradius
√2. Since the spectrum of non-hermitian random matrices from
the Ginibre ensemble lives asymptotically in a neighborhood of
the unitdisk, it follows that the outer belt of width
√2 − 1 containing no eigen-
values can be seen as a quantification the non-normality of the
complexGinibre random matrix. We also show that the numerical range
of uppertriangular Gaussian matrices converges to the same disk of
radius
√2,
while all eigenvalues are equal to zero and we prove that the
operatornorm of such matrices converges to
√2e.
1. INTRODUCTION
In this paper we are interested in the numerical range of large
randommatrices. In general, the numerical range (also called the
field of values) ofanN×N matrix is defined asW(X) = {(Xy, y) :
||y||2 = 1} (see e.g. [19,23, 25]). This notion was introduced
almost a century ago and it is known bythe celebrated
Toeplitz-Hausdorff theorem [22, 40] thatW(X) is a compactconvex set
in C. A common convention to denote the numerical range byW(X) goes
back to the German term “Wertevorrat” used by Hausdorff.
For any N × N matrix X its numerical range W(X) clearly contains
allits eigenvalues λi, i ≤ N. If X is normal, that is XX∗ = X∗X,
then its nu-merical range is equal to the convex hull of its
spectrum, W(X) = Γ(X) :=conv(λ1, . . . , λN). The converse is valid
if and only if N ≤ 4 ([34, 24]).
Date: February 28, 2014.1Research partially supported by ERA,
NSERC discovery grant, and AIMR.2Research partially supported by
the Grant N N516 481840 financed by Polish National
Centre of Science.3Research partially supported by the E.W.R.
Steacie Memorial Fellowship.4Research partially supported by the
Grant DEC-2011/02/A/ST1/00119 financed by
Polish National Centre of Science.1
arX
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4
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2 B. COLLINS, P. GAWRON, A. E. LITVAK, K. ŻYCZKOWSKI
For a non-normal matrix X its numerical range is typically
larger thanΓ(X) even in the case N = 2. For example, consider the
Jordan matrix oforder two,
J2 =
[0 10 0
].
Then both eigenvalues of J2 are equal to zero, while W(J2) forms
a diskD(0, 1/2).
We shall now turn our attention to numerical range of random
matrices.Let GN be a complex random matrix of order N from the
Ginibre ensem-ble, that is an N ×N matrix with i.i.d centered
complex normal entries ofvariance 1/N. It is known that the
limiting spectral distribution µN con-verges to the uniform
distribution on the unit disk with probability one (cf.[6, 16, 17,
18, 38, 39]). It is also known that the operator norm goes to 2with
probability one. This is directly related to the fact that the
level densityof the Wishart matrixGNG∗N is asymptotically described
by the Marchenko-Pastur law, supported on [0, 4], and the squared
largest singular value ofGNgoes to 4 ([20], see also [15] for the
real case).
As the complex Ginibre matrixGN is generically non-normal, the
supportΓ of its spectrum is typically smaller than the numerical
range W. Our re-sults imply that the ratio between the area ofW(GN)
and Γ(GN) convergesto 2 with probability one. Moreover, in the case
of strictly upper triangularmatrix TN with Gaussian entries (see
below for precise definitions) we havethat the area ofW(TN)
converges to 2, while clearly Γ(TN) = {0}.
The numerical range of a matrix X of size N can be considered as
aprojection of the set of density matrices of size N,
QN = {ρ : ρ = ρ∗, ρ ≥ 0, Trρ = 1},
onto a plane, where this projection is given by the (real)
linear map ρ 7→ TrρX.More precisely, for any matrix X of size N
there exists a real affine rank 2projection P of the setQN, whose
image is congruent to the numerical rangeW(X) [12].
Thus our results on numerical range of random matrices
contribute to theunderstanding of the geometry of the convex set of
quantum mixed statesfor large N.
Let dH denotes the Hausdorff distance. Our main result, Theorem
4.1,states the following:If random matrices XN of order N satisfy
for every real θ
limN→∞ ‖Re (eiθXN)‖ = R
then with probability one
limN→∞dH(W(XN), D(0, R)) = 0.
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NUMERICAL RANGE 3
We apply this theorem to a large class of random matrices.
Namely, letxi,i, i ≥ 1, be i.i.d. complex random variables with
finite second moment,xi,j, i 6= j, be i.i.d. centered complex
random variables with finite fourthmoment, and all these variables
are independent. Assume E|x1,2|2 = λ2 forsome λ > 0. Let XN =
N−1/2 {xi,j}i,j≤N, and YN be the matrix whose entriesabove the main
diagonal are the same as entries of XN and all other entriesare
zeros. Theorem 4.2 states that
dH(W(XN), D(0,√2λ)) → 0 and dH(W(YN), D(0, λ)) → 0.
In particular, if XN is a complex Ginibre matrix GN or a real
Ginibre matrixGRN (i.e. with centered normal entries of variance
1/N) and TN is a strictlytriangular matrices TN with i.i.d centered
complex normal entries of vari-ance 2/(N− 1) (so that ETrXNX∗N =
ETrTNT ∗N = N) then with probabilityone
dH(W(GN), D(0,√2)) → 0 and dH(W(TN), D(0,√2)) → 0.
We also provide corresponding quantitative estimates on the rate
of theconvergence in the case of GN and TN.
A related question to our study is the limit behavior of the
operator (spec-tral) norm ‖TN‖ of a random triangular matrix, which
can be used to char-acterize its non-normality. As we mentioned
above, it is known that withprobability one
(1) limN→∞ ‖GN‖ = 2.
It seems that the limit behavior of ‖TN‖ has not been
investigated yet, al-though its limiting counterpart has been
extensively studied by Dykema andHaagerup in the framework of
investigations around the invariant subspaceproblem. In the last
section (Theorem 6.2), we prove that with probabilityone
(2) limN→∞ ‖TN‖ =
√2e.
Note that in Section 6 this fact is formulated and proved in
another normal-ization.
Our proof here is quite indirect and relies on strong
convergence for ran-dom matrices established by [21]. In
particular, our proof does not pro-vide any quantitative estimates
for the rate of convergence. It would beinteresting to obtain
corresponding deviation inequalities. We would like tomention that
very recently the empirical eigenvalue measures for large classof
symmetric random matrices of the form XNX∗N, where XN is a
randomtriangular matrix, has been investigated ([31]).
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4 B. COLLINS, P. GAWRON, A. E. LITVAK, K. ŻYCZKOWSKI
The paper is organized as follows. In Section 2, we provide some
pre-liminaries and numerical illustrations. In Section 3, we
provide basic factson the numerical range and on the matrices
formed using Gaussian randomvariables. The main section, Section 4,
contains the results on convergenceof the numerical range of random
matrices mentioned above (and the cor-responding quantitative
estimates). Section 5 suggests a possible extensionof the main
theorem, dealing with a more general case, when the limit of‖Re
(eiθXN)‖ is a (non-constant) function of θ. Finally, in Section 6,
weprovide the proof of (2).
2. PRELIMINARIES AND NUMERICAL ILLUSTRATIONS
By ξ, we will denote a centered complex Gaussian random
variable,whose variance may change from line to line. When (the
variance of) ξis fixed, ξij, i, j ≥ 1 denote independent copies of
ξ. Similarly, by g wewill denote a centered real Gaussian random
variable, whose variance maychange from line to line. When (the
variance of) g is fixed, gij, i, j ≥ 1denote independent copies of
g.
We deal with random matrices XN of sizeN. To set the scale we
are usu-ally going to normalize random matrices by fixing their
expected Hilbert-Schmidt norms to be equal to N, i.e. E‖XN‖2HS =
ETrXNX∗N = N. Westudy the following ensembles.
(1) Complex Ginibre matrices GN of order N with entries ξij,
whereE|ξij|2 = 1/N. As we mention in the introduction, by the
circularlaw, the spectrum ofGN is asymptotically contained in the
unit disk.Note E‖GN‖2HS = N.
(2) Real Ginibre matricesGRN of orderNwith entries gij, where
E|gij|2 =1/N. Note E‖GRN‖2HS = N.
(3) Upper triangular random matrices TN of order N with entries
Tij =ξij for i < j and Tij = 0 elsewhere, where E|ξij|2 = 2/(N −
1).Clearly, all eigenvalues of TN equal to zero. Note E‖TN‖2HS =
N.
(4) Diagonalized Ginibre matrices, DN = ZGNZ−1 of order N, so
thatDkl = λkδkl where λk, k = 1, . . . ,N, denote complex
eigenval-ues of GN. Note that GN is diagonalizable with probability
one. Inorder to ensure the uniqueness of the probability
distribution on di-agonal matrices, we assume that it is invariant
under conjugation bypermutations. Note that integrating over the
Girko circular law onegets the average squared eigenvalue of the
complex Ginibre matrix,〈|λ|2〉 =
∫102x3dx = 1/2. Thus, E‖DN‖2HS = N/2.
(5) Diagonal unitary matricesUN of orderNwith entriesUkl =
exp(iφk)δkl,where φk are independent uniformly distributed on [0,
2π) real ran-dom variables.
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NUMERICAL RANGE 5
The structure of some of these matrices is exemplified below for
the caseN = 4. Note that the variances of ξ are different in the
case of G4 and inthe case of T4. To lighten the notation they are
depicted by the same symbolξ, but entries are independent.
G4 =
ξ ξ ξ ξξ ξ ξ ξξ ξ ξ ξξ ξ ξ ξ
, T4 =0 ξ ξ ξ0 0 ξ ξ0 0 0 ξ0 0 0 0
, D4 =λ1 0 0 00 λ2 0 00 0 λ3 00 0 0 λ4
.We will study the following parameters of a given (random)
matrix X:
(a) the numerical radius r(X) = max{|z| : z ∈W(X)},(b) the
spectral radius ρ(X) = |λmax|, where λmax is the leading
eigenvalue
of X with the largest modulus,(c) the operator (spectral) norm
equal to the largest singular value, ‖X‖ =
σmax(X) =√λmax(XX∗) (and equals to the operator norm of X,
con-
sidered as an operator `N2 → `N2 ),(d) the non-normality measure
µ3(X) := (||X||2HS −
∑Ni=1 |λi|
2)1/2.The latter quantity, used by Elsner and Paardekooper [14],
is based on the
Schur lemma: As the squared Hilbert-Schmidt norm of a matrix can
be ex-pressed by its singular values, ||X||2HS =
∑Ni=1 σ
2i , the measure µ3 quantifies
the difference between the average squared singular value and
the averagesquared absolute value of an eigenvalue, and vanishes
for normal matri-ces. Comparing the expectation values for the
squared norms of a randomGinibre matrix GN and a diagonal matrixDN
containing their spectrum weestablish the following statement.
The squared non-normality coefficient µ3 for a complex Ginibre
matrixGN behaves asymptotically as
(3) Eµ23(GN) = E‖GN‖2HS − E‖DN‖2HS = N/2.Since all eigenvalues
of random triangular matrices are equal to zero ananalogous results
for the ensemble of upper triangular random matricesreads Eµ23(TN)
= N.
Figure 1 shows the numerical range of the complex Ginibre
matrices ofensemble (1), which tends asymptotically to the disk of
radius
√2 – see
Theorem 4.2. As the convex hull of the spectrum, Γ(GN), goes to
the unitdisk, the ratio of their area tends to 2 and characterizes
the non-normalityof a generic Ginibre matrix. By the non-normality
belt we mean the setdifferenceW(X) \ Γ(X), which contains no
eigenvalues.
As N grows to infinity, spectral properties of the real Ginibre
matricesof ensemble (2) become analogous to the complex case. By
Theorem 4.2,
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6 B. COLLINS, P. GAWRON, A. E. LITVAK, K. ŻYCZKOWSKI
in both cases numerical range converges to D(0,√2) and the
spectrum is
supported by the unit disk. The only difference is the symmetry
of thespectrum with respect to the real axis and a clustering of
eigenvalues alongthe real axis for the real case.
−√
2−1 0 1√
2<
−√
2−1
0
1
√2
=
−√
2−1 0 1√
2<
−√
2−1
0
1
√2
=
−√
2−1 0 1√
2<
−√
2−1
0
1
√2
=
r
ρ
FIGURE 1. Spectrum (dots) and numerical range (dark con-vex set)
of the complex Ginibre matrices of sizes N =10, 100 and 1000. The
spectrum is asymptotically containedin the unit disk while
numerical range converges to a disk ofradius r =
√2 denoted in the figures. Note the outer ring
of the range is the non-normality belt of width√2 − 1 (it
contains no eigenvalues).
Figure 2 shows analogous examples of diagonal matrices D with
theGinibre spectrum – ensemble (4). Diagonal matrices are normal,
so the nu-merical range equals to the support of the spectrum and
thus converges tothe unit disk. Note that this property hold also
for a “normal Ginibre ensem-ble” of matrices of the kind G ′ =
VDV∗, where D contains the spectrumof a Ginibre matrix, while V is
a random unitary matrix drawn according tothe Haar measure.
Analogous results for the upper triangular matrices T of
ensemble (3)shown in Fig.3. The numerical range asymptotically
converges to the diskof radius
√2 with probability one – see Theorem 4.2.
As all eigenvalues of T are zero, the asymptotic properties of
the spec-trum and numerical range of T become identical with these
of a Jordanmatrix J of the same order N rescaled by
√2. By construction Jkm = 1 if
k + 1 = m and zero elsewhere for k,m = 1, . . . ,N. It is known
[41] thatnumerical range of a Jordan matrix J of sizeN converges to
the unit disk asN→ ∞.
In the table below we listed asymptotic predictions for the
operator (spec-tral) norm ‖X‖, the numerical radius r(X), the
spectral radius ρ(X) and thesquared non-normality parameter, µ̄23 =
E(µ23), of generic matrices pertain-ing to the ensembles
investigated.
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NUMERICAL RANGE 7
−√
2−1 0 1√
2<
−√
2−1
0
1
√2
=
−√
2−1 0 1√
2<
−√
2−1
0
1
√2
=
−√
2−1 0 1√
2<
−√
2−1
0
1
√2
= rρ
FIGURE 2. As in Fig. 1, for ensemble of diagonal matricesDN
containing spectrum of Ginibre matrices of sizes N =10, 100 and
1000. Numerical range of these normal matricescoincides with the
convex hull of their spectrum.
−√
2−1 0 1√
2<
−√
2−1
0
1
√2
=
−√
2−1 0 1√
2<
−√
2−1
0
1
√2
=
−√
2−1 0 1√
2<
−√
2−1
0
1
√2
=r
FIGURE 3. As in Fig. 1, for upper triangular random matri-ces TN
of sizes N = 10, 100 and 1000, for which all eigen-values are equal
to zero and the numerical range convergesto the disk of radius
√2.
Ensemble ‖X‖ r(X) ρ(X) µ̄23(X)Ginibre G 2
√2 1 N/2
Diagonal D 1 1 1 0
Triangular T√2e
√2 0 N
Consider a matrix X of orderN, normalized as TrXX∗ = N. Assume
thatthe matrix is diagonal, so that its numerical range W(X) is
formed by theconvex hull of the diagonal entries. Let us now modify
the matrix X, writingY =√1− aX+
√aT , where T is a strictly upper triangular random matrix
normalized as above and 0 ≤ a ≤ 1. Note TrYY∗ = N as well.
Rescaling Xby a number
√1− a smaller than one and adding an off-diagonal part
√aT
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8 B. COLLINS, P. GAWRON, A. E. LITVAK, K. ŻYCZKOWSKI
increases the non-normality belt of Y, i.e. the set W(Y) \ Γ(Y).
The largerrelative weight of the off-diagonal part, the larger
squared non-normalityindex, µ23(Y) = ‖Y‖2HS −
∑Ni=1 |Yii|
2 = N− (1− a)N = aN and the largerthe non-normality belt of the
numerical range. In the limiting case a → 1the off-diagonal
part
√aT dominates the matrix Y. In particular, if T = TN
of ensemble (3) then its numerical range converges to the disk
of radius√2
as N grows to infinity.To demonstrate this construction in
action we plotted in Fig. 4 numerical
range of an exemplary random matrix Y ′ = DN+ 1√2TN, which
contains thespectrum of the complex Ginibre matrix GN at the
diagonal, and the matrixTN in its upper triangular part. The
relative weight a = 1/
√2 is chosen
in such a way that TrY ′Y ′∗ = N. Thus Y ′ displays similar
properties tothe complex Ginibre matrix: its numerical range is
close to a disk of radiusr =√2, while the support of the spectrum
is close to the unit disk. This
observation is related to the fact [32] that bringing the
complex Ginibrematrix by a unitary rotation to its triangular Schur
form, S := UGU∗ =D+T , one assures that the diagonal matrixD
contains spectrum ofG, whileT is an upper triangular matrix
containing independent Gaussian randomnumbers.
−√
2−1 0 1√
2<
−√
2−1
0
1
√2
=
r
ρ
−2 −1 0 1 2<
−2
−10
1
2
=
r
ρ
FIGURE 4. As in Fig. 1, for a)DN + 1√2TN and b) UN + TNof size N
= 1000.
Another illustration of the non-normality belt is presented in
Fig. 4b. Itshows the numerical range of the sum of a diagonal
random unitary matrixUN of ensemble (5), with all eigenphases drawn
independently accordingto a uniform distribution, with the upper
triangular matrix TN of ensemble(3). All eigenvalues of this matrix
belong to the unit circle, while presenceof the triangular
contribution increases the numerical radius r and formsthe
non-normality belt. Some other examples of numerical range
computednumerically for various ensembles of random matrices can be
found in [36].
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NUMERICAL RANGE 9
3. SOME BASIC FACTS AND NOTATION
In this paper, C0, C1, ..., c1, c2, ... denote absolute positive
constants,whose value can change from line to line. Given a square
matrix X, wedenote
Re X =X+ X∗
2and Im X =
X− X∗
2i,
so that X = Re X+i Im X and both Re X and Im X are self-adjoint
matrices.Then it is easy to see that
ReW(X) =W(Re X) and ImW(X) =W(Im X).
Given θ ∈ [0, 2π], denote Xθ := eiθX and by λθ denote the
maximaleigenvalue of Re Xθ. It is known (see e.g. Theorem 1.5.12 in
[23]) that
(4) W(X) =⋂
0≤θ≤2π
Hθ,
whereHθ = e
−iθ {z ∈ C : Re z ≤ λθ} .Our results for random matrices are
somewhat similar, however we use thenorm ‖Xθ‖ instead of its
maximal eigenvalue. Repeating the proof of (4)(or adjusting the
proof of Proposition 5.1 below), it is not difficult to seethat
(5) W(X) ⊂ K(R),where K(R) is a star-shaped set defined by
(6) K(R) := {λe−iθ ‖Xθ‖ : λ ∈ [0, 1], θ ∈ [0, 2π)}.Below we
provide a complete proof of corresponding results for
randommatrices. Note that K(R) can be much larger thanW(X). Indeed,
in the caseof the identity operator I the numerical range is a
singleton, W(I) = {1},while the set K(R) is defined by the equation
ρ ≤ | cos t| (in the polarcoordinates).
3.1. GUE. We say that a HermitianN×N matrix A = {Ai,j}i,j
pertains toGaussian Unitary Ensemble (GUE) if a. its entries Ai,j’s
are independentfor 1 ≤ i ≤ j ≤ N, b. the entries Ai,j’s for 1 ≤ i
< j ≤ N are com-plex centered Gaussian random variables of
variance 1 (that is the real andimaginary parts are independent
centered Gaussian of variance 1/2), c. theentries Ai,i’s for 1 ≤ i
≤ N are real centered Gaussian random variables ofvariance 1.
Clearly, for the complex Ginibre matrix GN its real part, YN :=
Re(GN),is a (2N)−1/2 multiple of a GUE. It is known that with
probability one‖YN‖ → √2 (see e.g. Theorem 5.2 in [7] or Theorem
5.3.1 in [35]). We
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10 B. COLLINS, P. GAWRON, A. E. LITVAK, K. ŻYCZKOWSKI
will also need the following quantitative estimates. In [2, 27,
28, 30] it wasshown that for GUE, normalized as YN, one has for
every ε ∈ (0, 1],
P(‖YN‖ ≥
√2+ ε
)≤ C0 exp(−c0Nε3/2).
Moreover, in [30] it was also shown that for ε ∈ (0, 1],
P(‖YN‖ ≤
√2− ε
)≤ C1 exp(−c1N2ε3).
Note that C1 exp(−c1N2ε3) ≤ C2 exp(−c1Nε3/2). Thus, for ε ∈ (0,
1],
(7) P(|‖YN‖−
√2| > ε
)≤ C3 exp(−c2Nε3/2)
(cf. Theorem 2.7 in [10]). It is also well known (and follows
from concen-tration) that there exists two absolute constants c4
and C4 such that
(8) P (‖GN‖ ≥ 2.1) ≤ C4 exp(−c4N).
3.2. Upper triangular matrix. Let gi, hi, i ≥ 1, be independent
N(0, 1)real random variables. It is well-known (and follows from
the Laplace trans-form) that
Emaxi≤N
|gi| ≤√2 ln(2N).
Since ‖x‖∞ ≤ ‖x‖2, the classical Gaussian concentration
inequality (see[9] or inequality (2.35) in [26]) implies that for
every r > 0,
(9) P(
maxi≤N
|gi| >√2 ln(2N) + r
)≤ e−r2/2.
Recall that TN denotes the upper triangular N × N Gaussian
randommatrix normalized such that ETNT ∗N = N, that is (TN)ij are
independentcomplex Gaussian random variables of variance 2/(N− 1)
for 1 ≤ i < j ≤N and 0 otherwise. Note that Re TN can be
presented as ZN/
√2(N− 1),
where ZN is a complex Hermitian N×N matrix with zero on the
diagonaland independent complex Gaussian random variables of
variance one abovethe diagonal. Let AN be distributed as GUE (with
gi’s on the diagonal)and VN be the diagonal matrix with the same
diagonal as AN. Clearly,ZN = AN − VN. Therefore, the triangle
inequality and (7) yield that forevery ε ∈ (0, 1]
(10) P(∣∣∣∣ 1√N‖ZN‖− 2
∣∣∣∣ > ε) ≤ C exp(−cNε3/2),where C and c are absolute
positive constants (formally, applying the tri-angle inequality, we
should ask ε >
√ln(2N)/N, but if ε ≤
√ln(2N)/N
the right hand side becomes large than 1, by an appropriate
choice of the
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NUMERICAL RANGE 11
constant C). In particular, the Borel-Cantelli lemma implies
that with prob-ability one ‖ZN‖/
√N→ 2 (alternatively one can apply Theorem 5.2 from
[7]).
4. MAIN RESULTS
Our first main result is
Theorem 4.1. Let R > 0. Let {XN}N be a sequence of complex
randomN×N matrices such that for every θ ∈ R with probability
one
limN→∞ ‖Re (eiθXN)‖ = R.
Then with probability one
limN→∞dH (W(XN), D(0, R)) = 0.
Furthermore, if there exists A ≥ max{R, 1} such that for every N
≥ 1,P (‖XN‖ > A) ≤ pN
and for every ε ∈ (0, 1/2), N ≥ 1, θ ∈ R,P(∣∣ ‖Re (eiθXN)‖− R∣∣
> ε) ≤ qN(ε)
then for every positive ε ≤ min{1/2,√R/(A+ 1)} and every N one
has
P (dH(W(XN), D(0, R) > 4Aε) ≤ pN + 7Rε−2 qN(ε2).Proof. Fix
positive ε ≤ min{1/2, R/(A+1)}. Since the real part of a matrixis a
self-adjoint operator we have
λ(θ,N) := ‖Re (eiθXN)‖ = sup{Re (eiθXNy, y) : ‖y‖2 = 1}.By
assumptions of the theorem, for every θ ∈ R with probability
one
limN→∞ λ(θ,N) = R.
Let S denote the boundary of the disc D(0, R). Choose a finite
ε-net Nin [0, 2π], so that {Reiθ}θ∈N is an ε-net (in the geodesic
metric) in S. Then,with probability one, for every θ ∈ N one has
λ(θ,N) → R.
Since Im XN = Re (e−iπ/2XN), one has
R ≤ lim supN→∞ ‖XN‖ ≤ lim supN→∞ ‖Re XN‖+ lim supN→∞ ‖Im XN‖ =
2R.
Choose A ≥ max{R, 1} and N ≥ 1 such that for everyM ≥ N one
has‖XM‖ ≤ A and ∀θ ∈ N |λ(θ,M) − R| ≤ ε.
FixM ≥ N. Note that the supremum in the definition of λ(θ,M) is
attainedand that
|Re (eiθXMy, y) − Re (eitXMy, y)| ≤ |eiθ − eit| · |(XMy, y)| ≤
εA,
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12 B. COLLINS, P. GAWRON, A. E. LITVAK, K. ŻYCZKOWSKI
whenever |θ − t| ≤ ε and ‖y‖2 = 1. Using approximation by
elements ofN, we obtain for every real t,
|λ(t,M) − R| ≤ (A+ 1)ε.Let y0 be such that ‖y0‖ = 1 and
λ := sup{|(XMy, y)| : ||y||2 = 1} = |(XMy0, y0)|.
Then for some t
λ = eit(XMy0, y0) = Re (eitXMy0, y0) = λ(t,M) ≤ R+ (A+ 1)ε.This
shows thatW(XM) ⊂ D(0, R+ (A+ 1)ε).
Finally fix some z ∈ S, that is z = Reit. Choose θ ∈ N such
that|t− θ| ≤ ε. Let y1 be such that
λ(−θ,M) = Re (e−iθXMy1, y1) = Re (e−iθ(XMy1, y1)).
Denote x := (XNy1, y1). Then
R− (A+ 1)ε ≤ Re (e−iθx) ≤ |x| ≤ R+ (A+ 1)ε.Since A ≥ max{R, 1}
and ε ≤ R/(A+ 1), this implies that
|Reiθ − x| ≤√
(A+ 1)2ε2 + 4R(A+ 1)ε ≤ 2A√ε√ε+ 2.
Since |t− θ| ≤ ε and ε < 1/2, we observe that|z− x| ≤ R|eit −
eiθ|+ |Reiθ − x| ≤ Rε+ 2
√2.5A
√ε ≤ 4A
√ε.
Therefore, for every z ∈ S there exists x ∈W(XM) with|z− x| ≤
4A
√ε.
Using convexity ofW(XM), we obtain that with probability one
dH(W(XM), D(0, R)) ≤ 4A√ε.
SinceM ≥ N was arbitrary, this implies the desired result.The
proof of the second part of the theorem is essentially the same.
Note
that the ε-net in our proof can be chosen to have the
cardinality not exceed-ing 2.2πR/ε. Thus, by the union bound, the
probability of the event
‖XM‖ ≤ A and ∀θ ∈ N |λ(θ,M) − R| ≤ ε,considered above, does not
exceed pN + 2.2πR ε−1qN(ε). This implies thequantitative part of
the theorem. �
The next theorem shows that the first part of Theorem 4.1
applies to alarge class of random matrices (essentially to matrices
whose entries arei.i.d. random variables having final fourth
moments and corresponding tri-angular matrices), in particular to
ensembles GN, GRN and TN introduced inSection 2.
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NUMERICAL RANGE 13
Theorem 4.2. Let xi,i, i ≥ 1, be i.i.d. complex random variables
with finitesecond moment, xi,j, i 6= j, be i.i.d. centered complex
random variableswith finite fourth moment, and all these variables
are independent. AssumeE|x1,2|2 = λ2 for some λ > 0. Let XN =
N−1/2 {xi,j}i,j≤N, and YN be thematrix whose entries on or above
the diagonal are the same as entries ofXN and entries below
diagonal are zeros. Then with probability one,
dH(W(XN), D(0,√2λ)) → 0 and dH(W(YN), D(0, λ)) → 0.
In particular with probability one,
dH(W(GN), D(0,√2)) → 0, dH(W(GRN), D(0,√2)) → 0
anddH(W(TN), D(0,
√2)) → 0.
Proof. It is easy to check that the entries of√NRe(eiθXN)
satisfy condi-
tions of Theorem 5.2 in [7], that is the diagonal entries are
i.i.d. real randomvariables with finite second moment; the above
diagonal entries are i.i.d.mean zero complex variables with finite
fourth moment and of varianceλ2/2. Therefore, Theorem 5.2 in [7]
implies that ‖Re(eiθXN)‖ → √2λ.Theorem 4.1 applied with R =
√2λ provides the first limit. For the trian-
gular matrix YN the proof is the same, we just need to note that
the abovediagonal entries of
√NRe(eiθYN) have variances (λ/2)2. The “in particu-
lar” part follows immediately. �
We now turn to quantitative estimates for ensembles GN and
TN.
Theorem 4.3. There exist absolute positive constants c and C
such that forevery ε ∈ (0, 1] and every N,
P(dH
(W(GN), D(0,
√2))≥ ε)≤ C ε−2 exp(−cNε3).
Remark 1. Note that by Borel-Cantelli lemma, this theorem also
impliesthat dH(W(GN), D(0,
√2)) → 0.
Proof. Note that for every real θ the distributions ofGN and
eiθGN coincide.Note also that Re (GN) is a 1/
√2N multiple of a GUE. Thus, the desired
result follows from the quantitative part of Theorem 4.1 by (7)
and (8) (andby adjusting absolute constants). �
Remark 2. It is possible to establish a direct link between
Theorem 4.3,geometry of the set of mixed quantum states and the
Dvoretzky theorem[11, 33].
-
14 B. COLLINS, P. GAWRON, A. E. LITVAK, K. ŻYCZKOWSKI
As before, let QN = {ρ : ρ = ρ∗, ρ ≥ 0, Trρ = 1} be the set
ofcomplex density matrices of size N. It is well known [8] that
workingin the geometry induced by the Hilbert-Schmidt distance this
set of (real)dimension N2 − 1 is inscribed inside a sphere of
radius
√(N− 1)/N ≈
1, and it contains a ball of radius 1/√(N− 1)N ≈ 1/N. Applying
the
Dvoretzky theorem and the techniques of [4], one can prove the
followingresult [5]: for largeN a generic two-dimensional
projection of the setQN isvery close to the Euclidean disk of
radius rN = 2/
√N. Loosely speaking,
in high dimensions a typical projection of a convex body becomes
close toa circular disk – see e.g. [3].
To demonstrate a relation with the numerical range of random
matri-ces we apply results from [12], where it was shown that for
any matrixX of order N its numerical range W(X) is up to a
translation and dilationequal to an orthogonal projection of the
set QN. The matrix X determinesthe projection plane, while the
scaling factor for a traceless matrix reads
α(X) =√
12(TrXX∗ + |TrX2|).
Complex Ginibre matrices are asymptotically traceless, and the
secondterm |TrG2| tends to zero, so the normalization condition
used in this work,ETrGNG∗N = N, implies that Eα(GN) converges
asymptotically to
√N/2.
It is natural to expect that the projection of QN associated
with the com-plex Ginibre matrix GN is generic and is characterized
by the Dvoretzkytheorem.
Our result shows that the random projection of QN, associated
with thecomplex Ginibre matrix GN does indeed have the features
expected in viewof Dvoretzky’s theorem and is close to a disk of
radius rNEα(GN) =
√2.
Theorem 4.4. There exist absolute positive constants c and C
such that forevery ε ∈ (0, 1] and every N,
P(dH
(W(TN), D(0,
√2))≥ ε)≤ C ε−2 exp(−cNε3).
Remark 3. Note that by Borel-Cantelli lemma, this theorem also
impliesthat dH(W(TN), D(0,
√2)) → 0.
Proof. Note that for every real θ the distributions of TN and
eiθTN coincide.As was mentioned above Re TN can be presented as
ZN/
√2(N− 1), where
ZN is a complex Hermitian N × N matrix with zero on the diagonal
andindependent complex Gaussian random variables of variance one
above thediagonal. Thus, by (10), for every θ ∈ R and ε ∈ (0,
1]
P(∣∣∣‖Re (eiθTN)‖−√2∣∣∣ > ε) ≤ C exp(−cNε3/2)
-
NUMERICAL RANGE 15
(one needs to adjust the absolute constants). SinceXN = Re
XN+iIm XN =Re XN + iRe (e−iπ/2XN),
P (‖TN‖ ≥ 3) ≤ C2 exp(−c1N).Thus, applying Theorem 4.1 (with R
=
√2 and A = 3), we obtain the
desired result. �
5. FURTHER EXTENSIONS.
Note that the first part of the proof of Theorem 4.1, the
inclusion ofW(XN) into the disk, can be extended to a more general
setting, when Ris not a constant but a function of θ. Namely, let R
: R → [1,∞) be a(2π)-periodic continuous function. Let K(R) be
defined by (6), i.e.
K(R) := {λe−iθ R(θ) : λ ∈ [0, 1], θ ∈ [0, 2π)}.Note that if we
identify C with R2 and θ with the direction e−iθ then Rbecomes the
radial function of the star-shaped body K(R). Then we havethe
following
Theorem 5.1. Let K(R) be a star-shaped body with a continuous
radialfunction R(θ), θ ∈ [0, 2π). Let {XN}N be a sequence of
complex randomN×N matrices such that for every θ ∈ [0, 2π) with
probability one
limN→∞ ‖Re (eiθXN)‖ = R(θ).
Then with probability one
limN→∞dH(W(XN) \ K(R), ∅) = 0
(in other words asymptotically the numerical range is contained
in K(R)).Furthermore, if there exists A > 0 such that for every
N ≥ 1,
P (‖XN‖ > A) ≤ pNand for every ε ∈ (0, 1/2), N ≥ 1, θ ∈
R,
P(∣∣ ‖Re (eiθXN)‖− R(θ)∣∣ > ε) ≤ qN(ε)
then for every ε ∈ (0, 1/2) and every N one hasP (dH(W(XN) ⊂
K(R+ (2A+ 1)ε)) ≤ pN + 2Lε−1 qN(ε),
where L denotes the length of the curve {R(θ)}θ∈[0,2π).
Remark 4. The proof below can be adjusted to prove the inclusion
(5) (infact (5) is simpler, since it does not require the
approximation).
-
16 B. COLLINS, P. GAWRON, A. E. LITVAK, K. ŻYCZKOWSKI
Remark 5. Under assumptions of Proposition 5.1 on the
convergence ofnorms to R, the function R must be continuous.
Indeed, for every θ and tone has with probability one
|R(θ) − R(t)| ≤ limN→∞
∣∣ ‖Re (eiθXN)‖− ‖Re (eitXN)‖∣∣≤∣∣eiθ − eit∣∣ lim sup
N→∞ ‖XN‖and
lim supN→∞ ‖XN‖ ≤ lim supN→∞ ‖Re XN‖+ lim supN→∞ ‖Im XN‖ = R(0)
+ R(−π/2).
Remark 6. Continuity and periodicity are not the only
constraints that Rshould satisfy. For Theorem 5.1 not to be an
empty statement, The set K(R)should also have the property of being
convex. This is clearly a necessarycondition, and it can be proved
by simple diagonal examples that it is alsoa sufficient
condition.
Proof. Fix ε ∈ (0, 1/2). Denoteλ(θ,N) := sup{Re (eiθXNy, y),
‖y‖2 = 1}.
Note thatλ(θ,N) = ‖Re (eiθXN)‖.
Thus for every θ ∈ R with probability onelimN→∞ λ(θ,N) =
R(θ).
Let ∂K = {R(θ) | θ ∈ [0, 2π)} denote the boundary of K(R).
Choose afinite set N in [0, 2π] so that {R(θ)eiθ}θ∈N is an ε-net in
∂K (in the Euclideanmetric). Then, with probability one, for every
θ ∈ N one has λ(θ,N) →R(θ).
As before, note
maxθR(θ) ≤ lim sup
N→∞ ‖XN‖ ≤ R(0) + R(−π/2).Choose A ≥ 1 and N ≥ 1 such that for
everyM ≥ N one has
‖XM‖ ≤ A and ∀θ ∈ N |λ(θ,M) − R(θ)| ≤ ε.Note that the supremum
in the definition of λ(θ,N) is attained and that
|eiθ − eit| ‖XN‖ ≤ |eiθ − eit|A,whenever ‖y‖2 = 1. As was
mentioned in the remark following the theo-rem,
|R(θ) − R(t)| ≤∣∣eiθ − eit∣∣A.
-
NUMERICAL RANGE 17
Therefore, using approximation by elements of N and the simple
estimate∣∣eiθ − eit∣∣ ≤ ε, whenever |θ − t| ≤ ε, we obtain that for
every real t onehas
|λ(t,N) − R(t)|
≤ |λ(t,N) − λ(θ,N)|+ |λ(θ,N) − R(θ)|+ |R(θ) − R(t)|≤ (2A+
1)ε.(11)
Now let y0 of norm one be such that (XNy0, y0) is in the
direction eit,that is (XNy0, y0) = eitR for some real positive R.
Then
R = e−it(XNy0, y0) = Re (e−itXNy0, y0) ≤ λ(−t,N) ≤
R(−t)+(2A+1)ε.This shows thatW(XN) ⊂ K(R+ (2A+ 1)ε).
The quantitative estimates are obtained in the same way as in
the proofof Theorem 4.1. �
As an example consider the following matrix. LetH1,H2 be
independentdistributed asGN, a, b > 0 andA := aH1+ibH2. Then it
is easy to see thatRe (eiθA) is distributed as r(θ)GN, where r(θ)
=
√a2 cos2 θ+ b2 sin2 θ.
Therefore ‖Re (eiθA)‖ → R(θ) := √2r(θ). Theorem 5.1 implies
thatW(A) is asymptotically contained in K(R) which is an
ellipse.
6. NORM ESTIMATE FOR THE UPPER TRIANGULAR MATRIX
In this section we prove that ‖TN‖ → √2e, as claimed in Eq. (2)
of theintroduction (Theorem 6.2). For the purpose of this section
it is convenientto renormalize the matrix TN and to consider T̄N,
which is strictly upperdiagonal and whose entries above the
diagonal are complex centered i.i.d.Gaussians of variance 1/
√N. Thus, (T̄N)ij =
√(N− 1)/(2N)Tij.
We also consider upper triangular matrices T ′N, whose entries
above andon the diagonal are complex centered i.i.d. Gaussians of
variance 1/
√N.
Note that T̄N and T ′N differs on the diagonal only, therefore
the followinglemma follows from (9).
Lemma 6.1. The operator norm of T̄N converges with probability
one to alimit L iff the operator norm of T ′N converges with
probability one to L.
We reformulate the limiting behavior of ‖TN‖ in terms of T̄N. We
provethe following theorem, which is clearly equivalent to (2).
Theorem 6.2. With probability one, the operator norm of the
sequence ofrandom matrices T̄N tends to
√e.
Let us first recall the following theorem, proved in [13].
-
18 B. COLLINS, P. GAWRON, A. E. LITVAK, K. ŻYCZKOWSKI
Proposition 6.3. For any integer `,
limN
E(N−1Tr((T̄NT̄ ∗N)`) =``
(`+ 1)!
We will use the following auxiliary constructions. Fix a
positive integerparameter k, denote m = [N/k] (the largest integer
not exceeding N/k),and define the upper triangular matrix T̄N,k as
follows: (T̄N,k)i,j = 0 if `m+1 ≤ j ≤ (` + 1)m and i ≥ `m + 1 for
some ` ≥ 0, and (T̄N,k)i,j = (T̄N)i,jotherwise. In other words we
set more entries to be equal to 0 and we haveeither k × k or (k +
1) × (k + 1) block strictly triangular matrix (if N isnot multiple
of k then the last, (k+ 1)th, “block-row” and “block-column”have
either their number of rows or columns strictly less than N/k).
We start with the following
Lemma 6.4. Let k be a positive integer andN be a multiple of k.
Then withprobability one, ‖T̄N,k‖ converges to a quantity fk as N→
∞.Proof. Note that the complex Ginibre matrix is, up to a proper
normaliza-tion, distributed asA1+iA2, whereA1 andA2 are i.i.d. GUE.
Thus, whenNis a multiple of k, T̄N,k can be seen as a k×k block
matrix ofN/k×N/kma-trices, which are linear combinations of i.i.d.
copies of GUE. A Haagerup-Thorbjornsen result [21] ensures
convergence with probability one of thenorm. �
At this point it is not possible to compute fk explicitly.
Actually it willbe enough for us to understand the asymptotics of
fk as k→ ∞.
In the next lemma, we remove the condition that N be a multiple
of k.
Lemma 6.5. Let k be a positive integer. Then with probability
one, ‖T̄N,k‖converges to to the quantity fk defined in Lemma 6.4 as
N→ ∞.Proof. Let N ≥ k. Denote by N+ the first multiple of k after
N. Up toan overall multipleN/N+ (imposed by the normalization that
is dimensiondependent), we can realize T̄N,k as a compression of
T̄N+,k. Since a com-pression reduces the operator norm, thanks to
the previous lemma, we havewith probability one,
lim supN→∞ ‖T̄N,k‖ ≤ fk.
Similarly, by N− denote the first multiple of k before N. Up to
an overallmultiple N−/N, we can realize T̄N−,k as a compression of
T̄N,k. Thereforewe have with probability one,
lim infN→∞ ‖T̄N,k‖ ≥ fk.
These two estimates imply the lemma. �
-
NUMERICAL RANGE 19
In the next Lemma, we compare the norm of T̄N,k with the norm of
T̄N.
Lemma 6.6. With probability one for every k we have
lim supN→∞
∣∣‖T̄N,k‖− ‖T̄N‖∣∣ ≤ 3/√k.Proof. For every fixed k ≤ N we
consider a matrix DN,k distributed asT̄N,k − T̄N. Setting as before
m = [N/k], the entries of DN,k are i.i.d.Gaussian of variance
1/
√N if `m+ 1 ≤ j ≤ (`+ 1)m and i ≥ `m+ 1 for
some ` ≥ 0, and (DN,k)i,j = 0 otherwise. Clearly, this matrix is
diagonal byblock. It consists of k diagonal blocks of m ×m strictly
upper triangularrandom matrices with entries of variance 1/N and
possibly one more blockof smaller size.
Let us first work on estimating the tail of the operator norm on
a diagonalblock of sizem×m, which will be denoted by XN. It follows
directly fromthe Wick formula that the quantities E(Tr((XNX∗N)`))
are bounded aboveby quantities E(Tr((X̃NX̃∗N)`)), where X̃N is the
same matrix as XN with-out the assumption that lower triangular
entries are zero (in other words,it is a rescaled complex Ginibre
matrix of size m × m). From there,we can make estimates following
arguments à la Soshnikov [37] and ob-tain that the tail of the
operator norm of XN is majorized by the tail ofthe operator norm of
X̃N. More precisely, we can show that there existsa constant C1
> 0 such that E(Tr((XNX∗N)`)) ≤ C1(2.8/
√k)` for every
` ≤ N1/4. This implies that there exists another constant C2
> 0 such thatE(Tr(D`N,k)) ≤ C1k(2.8/
√k)` ≤ C2(2.9/
√k)` for all sufficiently large
` ≤ N1/4. Therefore we deduce by Jensen inequality that the
probabilitythat the operator norm DN,k is larger than 3/
√k is bounded by C−N for
some universal constant C > 1. By Borel-Cantelli lemma, with
probabilityone we have
lim supN→∞
∣∣‖T̄N,k‖− ‖T̄N‖∣∣ ≤ 3/√k.The result follows by the triangle
inequality. �
As a consequence we obtain the following lemma.
Lemma 6.7. The sequence fk converges to some constant f as k→ ∞
and‖T̄N‖ converges to f with probability one.Proof. By Lemma 6.6
and the triangle inequality, we get that with proba-bility one,
lim supN→∞
∣∣‖T̄N,k1‖− ‖T̄N,k2‖∣∣ ≤ 3/√k1 + 3/√k2.Therefore, evaluating the
limit on the left hand side, we observe that {fk}kis a Cauchy
sequence. Thus it converges to a constant f.
-
20 B. COLLINS, P. GAWRON, A. E. LITVAK, K. ŻYCZKOWSKI
Next, we see that for any ε > 0, taking k large enough, we
obtain thatwith probability one,
lim supN→∞
∣∣‖T̄N‖− f∣∣ ≤ ε.Letting ε→ 0, we obtain the desired result.
�
Now we are ready to finish the proof of Theorem 6.2.
Proof of Theorem 6.2. It is enough to prove that f =√e. It
follows from
[21] that
fk = lim`→∞ 2
√̀limN→∞E(N−1Tr((T̄N,kT̄ ∗N,k)`)).
Given ` andN, it follows from Wick’s theorem that
E(N−1Tr((T̄N,kT̄ ∗N,k)`))increases and converges as k→ ∞
pointwisely to E(N−1Tr((T̄NT̄ ∗N)`)). Sothe same result holds if we
let N→ ∞ (by Dini’s theorem), namely
limk→∞ limN→∞E(N−1Tr((T̄N,kT̄ ∗N,k)`)) = limN→∞E(N−1Tr((T̄NT̄
∗N)`)).
Observing that
2
√̀limN
E(N−1Tr((T̄N,kT̄ ∗N,k)`))
increases as a function of ` and applying once more Dini’s
theorem, weobtain that
limk→∞ fk = lim`→∞ 2
√̀limN→∞E(N−1Tr((T̄NT̄ ∗N)`)).
Therefore
limk→∞ fk = lim`→∞ 2
√̀``
(`+ 1)!=√e
by the Stirling formula. This completes the proof. �
Acknowledgment. We are grateful to Guillaume Aubrun and
StanisławSzarek for fruitful discussions on the geometry of the set
of quantum states,helpful remarks, and for letting us know about
their results prior to publi-cation. It is also a pleasure to thank
Zbigniew Puchała and Piotr Śniady foruseful comments. Finally we
would like to thank an anonymous referee forcareful reading and
valuable remarks which have helped us to improve
thepresentation.
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NUMERICAL RANGE 21
REFERENCES
[1] G. W. Anderson, A. Guionnet, O. Zeitouni, An introduction to
random matrices,Cambridge Studies in Advanced Mathematics, 118.
Cambridge University Press,Cambridge, 2010.
[2] G. Aubrun, A sharp small deviation inequality for the
largest eigenvalue of a randommatrix, Seminaire de Probabilites
XXXVIII, 320–337, Lecture Notes in Math., 1857,Springer, Berlin,
2005.
[3] G. Aubrun and J. Leys, Quand les cubes deviennent
ronds,http://images.math.cnrs.fr/Quand-les-cubes-deviennent-ronds.html
[4] G. Aubrun, S. J. Szarek, Tensor products of convex sets and
the volume of separablestates on N qudits, Phys. Rev. A 73 022109
(2006).
[5] G. Aubrun, S. J. Szarek, private communication[6] Z. D. Bai,
Circular law, Ann. Probab. 25 (1997), 494–529.[7] Z. D. Bai, J. W.
Silverstein, Spectral analysis of large dimensional random
matrices.
Second edition. Springer Series in Statistics. Springer, New
York, 2010.[8] I. Bengtsson and K. Życzkowski, Geometry of Quantum
States, Cambridge Univer-
sity Press, Cambridge, 2006.[9] B. S. Cirel’son, I. A.
Ibragimov, V. N. Sudakov, Norms of Gaussian sample functions,
Proc. 3rd Japan-USSR Symp. Probab. Theory, Taschkent 1975, Lect.
Notes Math. 550(1976), 20–41.
[10] K. R. Davidson, S. J. Szarek, Local operator theory, random
matrices and Banachspaces, Handbook of the geometry of Banach
spaces, Vol. I, 317–366, North-Holland,Amsterdam, 2001.
[11] A. Dvoretzky, Some results on convex bodies and Banach
spaces, Proc. Internat.Sympos. Linear Spaces, Jerusalem 1960; p.
123–160 (1961).
[12] C. F. Dunkl, P. Gawron, J. A. Holbrook, J. Miszczak, Z.
Puchała, K. Życzkowski,Numerical shadow and geometry of quantum
states, J. Phys. A 44 (2011) 335301 (19pp.)
[13] K. Dykema, U. Haagerup, DT-operators and decomposability of
Voiculescu’s circularoperator, Amer. J. Math. 126 (2004),
121–189.
[14] L. Elsner and M. H. C. Paardekooper, On measures of
nonnormality of matrices, Lin.Alg. Appl. 92, 107–124 (1987).
[15] S. Geman, A limit theorem for the norm of random matrices,
Ann. Probab. 8 (1980),252–261.
[16] J. Ginibre, Statistical ensembles of complex, quaternion,
and real matrices, J. Math.Phys. 6 (1965), 440–449.
[17] V. L. Girko, The circular law, Teor. Veroyatnost. i
Primenen. 29 (1984), 669–679.[18] F. Götze, A. Tikhomirov, The
circular law for random matrices, Ann. Probab. 38
(2010), 1444–1491.[19] K. E. Gustafson and D. K. M. Rao,
Numerical Range: The Field of Values of Linear
Operators and Matrices. Springer-Verlag, New York, 1997.[20] U.
Haagerup and S. Thorbjørnsen, Random matrices with complex Gaussian
entries.
Expositiones Math. 21 (2003), 293–337.[21] U. Haagerup, S.
Thorbjørnsen, A new application of random matrices:
Ext(C∗red(F2))
is not a group, Ann. of Math. 162 (2005), 711–775.[22] F.
Hausdorff, Der Wertevorrat einer Bilinearform, Math. Zeitschrift 3
(1919), 314–
316.
http://images.math.cnrs.fr/Quand-les-cubes-deviennent-ronds.html
-
22 B. COLLINS, P. GAWRON, A. E. LITVAK, K. ŻYCZKOWSKI
[23] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis,
Cambridge UniversityPress, Cambridge, 1994.
[24] C. R. Johnson, Normality and the Numerical Range, Lin. Alg.
Appl. 15, 89–94(1976).
[25] R. Kippenhahn, Über den Wertevorrat einer Matrix, Math.
Nachr. 6193–228 (1951).[26] M. Ledoux, The concentration of measure
phenomenon, Mathematical Surveys and
Monographs, 89. American Mathematical Society, Providence, RI,
2001.[27] M. Ledoux, A remark on hypercontractivity and tail
inequalities for the largest eigen-
values of random matrices. Seminaire de Probabilites XXXVII,
360–369, LectureNotes in Math., 1832, Springer, Berlin, 2003.
[28] M. Ledoux, Differential operators and spectral
distributions of invariant ensemblesfrom the classical orthogonal
polynomials. The continuous case, Electron. J. Probab.9 (2004), no.
7, 177–208
[29] M. Ledoux, A recursion formula for the moments of the
Gaussian orthogonal ensem-ble, Ann. Inst. Henri Poincare Probab.
Stat. 45 (2009), no. 3, 754–769.
[30] M. Ledoux, B. Rider, Small deviations for beta ensembles,
Electron. J. Probab. 15(2010), 1319–1343.
[31] A. Lytova, L. Pastur, On a Limiting Distribution of
Singular Values of Random Tri-angular Matrices, preprint.
[32] M. L. Mehta, Random Matrices, Elsevier, Amsterdam,
2004.[33] V. D. Milman, G. Schechtman, Asymptotic theory of
finite-dimensional normed
spaces. With an appendix by M. Gromov. Lecture Notes in
Mathematics, 1200.Springer-Verlag, Berlin, 1986.
[34] B. N. Moyls, M.D. Marcus, Field convexity of a square
matrix, Proc. Amer. Math.Soc. 6 (1955), 981–983.
[35] L. Pastur, M. Shcherbina, Eigenvalue distribution of large
random matrices. Mathe-matical Surveys and Monographs, 171.
American Mathematical Society, Providence,RI, 2011.
[36] Ł. Pawela et al., Numerical Shadow, The web resource at
http://numericalshadow.org[37] A. Soshnikov Universality at the
edge of the spectrum in Wigner random matrices,
Commun. Math. Phys. 207 (1999), 697–733.[38] T. Tao, V.H. Vu,
Random matrices: the circular law, Commun. Contemp. Math. 10
(2008), 261–307.[39] T. Tao, V.H. Vu, Random matrices:
Universality of ESD and the Circular Law (with
appendix by M. Krishnapur), Ann. of Prob. 38 (2010),
2023–2065.[40] O. Toeplitz, Das algebraische Analogon zu einem
Satze von Fejér, Math. Zeitschrift
2 (1918), 187–197.[41] P.-Y. Wu, A numerical range
characterization of Jordan blocks, Linear and Multilin-
ear Algebra, 43 (1998), 351–361.
DÉPARTEMENT DE MATHÉMATIQUE ET STATISTIQUE, UNIVERSITÉ
D’OTTAWA, 585KING EDWARD, OTTAWA, ON, K1N6N5 CANADA, WPI ADVANCED
INSTITUTE FORMATERIALS RESEARCH TOHOKU UNIVERSITY, MATHEMATICS UNIT
2-1-1 KATAHIRA,AOBA-KU, SENDAI, 980-8577 JAPAN AND CNRS, INSTITUT
CAMILLE JORDAN UNI-VERSITÉ LYON 1, FRANCE
E-mail address: [email protected]
http://numericalshadow.org
-
NUMERICAL RANGE 23
INSTITUTE OF THEORETICAL AND APPLIED INFORMATICS, POLISH ACADEMY
OFSCIENCES, BAŁTYCKA 5, 44-100 GLIWICE, POLAND
E-mail address: [email protected]
DEPT. OF MATH. AND STAT. SCIENCES, UNIVERSITY OF ALBERTA,
EDMONTON,ALBERTA, CANADA, T6G 2G1,
E-mail address: [email protected]
INSTITUTE OF PHYSICS, JAGIELLONIAN UNIVERSITY, UL. REYMONTA 4,
30-059KRAKÓW, POLAND
E-mail address: [email protected]
CENTER FOR THEORETICAL PHYSICS, POLISH ACADEMY OF SCIENCES, AL.
LOT-NIKÓW 32/46, 02-668 WARSZAWA, POLAND
1. Introduction2. Preliminaries and numerical illustrations3.
Some basic facts and notation3.1. GUE3.2. Upper triangular
matrix
4. Main results5. Further extensions.6. Norm estimate for the
upper triangular matrixReferences